We consider the minimum spanning tree problem in a setting where

information about the edge weights of the given graph is uncertain.

Initially, for each edge $e$ of the graph only a set $A_e$, called

an uncertainty area, that contains the actual edge weight

$w_e$ is known. The algorithm can `update' $e$ to obtain the edge

weight $w_e in A_e$. The task is to output the edge set of a

minimum spanning tree after a minimum number of updates. An

algorithm is $k$-update competitive if it makes at most $k$ times

as many updates as the optimum. We present a $2$-update

competitive algorithm if all areas $A_e$ are open or trivial, which

is the best possible among deterministic algorithms. The condition

on the areas $A_e$ is to exclude degenerate inputs for which no

constant update competitive algorithm can exist.

Next, we consider a setting where the vertices of the graph

correspond to points in Euclidean space and the weight of an edge

is equal to the distance of its endpoints. The location of each

point is initially given as an uncertainty area, and an update

reveals the exact location of the point. We give a general

relation between the edge uncertainty and the vertex uncertainty

versions of a problem and use it to derive a $4$-update competitive

algorithm for the minimum spanning tree problem in the vertex

uncertainty model. Again, we show that this is best possible among

deterministic algorithms.